JEE Main & Advanced Mathematics Definite Integration Question Bank

done Mock Test - Application of Integrals Total Questions - 25

• question_answer1)
  The area of the closed figure bounded by \[x=-1,\]\[y=0,\] \[y={{x}^{2}}+x+1\], and the tangent to the curve \[y={{x}^{2}}+x+1\] at A(1,3) is
  A)
  4/3 sq. units       done clear
  B)
  7/3 sq. units done clear
  C)
  7/6 sq. units       done clear
  D)
  None of these done clear
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• question_answer2)
  The area of the closed figure bounded \[y=\frac{{{x}^{2}}}{2}-2x+2\] and the tangents to it at (1, 1/2) and (4, 2) is
  A)
  9/8 sq. units       done clear
  B)
  3/8 sq. units done clear
  C)
  3/2 sq. units       done clear
  D)
  9/4 sq. units done clear
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• question_answer3)
  The area bounded by the two branches of curve \[{{(y-x)}^{2}}={{x}^{3}}\] and the straight line x = 1 is
  A)
  1/5 sq. units       done clear
  B)
  3/5 sq. units done clear
  C)
  4/5 sq. units       done clear
  D)
  8/4 sq. units done clear
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• question_answer4)
  The area of the region bounded by \[{{x}^{2}}+{{y}^{2}}-2x-3=0\] and \[y=\left| x \right|+1\]is
  A)
  \[\frac{\pi }{2}-1\] sq. units done clear
  B)
  \[2\pi \]sq. units done clear
  C)
  \[4\pi \]sq. units  done clear
  D)
  \[\pi /2\]sq. units done clear
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• question_answer5)
  The area of the loop of the curve \[a{{y}^{2}}={{x}^{2}}(a-x)\] is
  A)
  \[4{{a}^{2}}\]sq. units  done clear
  B)
  \[\frac{8{{a}^{2}}}{15}\]sq. units done clear
  C)
  \[\frac{16{{a}^{2}}}{9}\]sq. units done clear
  D)
  None of these done clear
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• question_answer6)
  The area of the figure bounded by the parabola \[{{(y-2)}^{2}}=x-1\], the tangent to it at the point with the ordinate x = 3, and the x-axis is
  A)
  7 sq. units          done clear
  B)
  6 sq. units done clear
  C)
  9 sq. units          done clear
  D)
  None of these done clear
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• question_answer7)
  The area enclosed by the curve \[y=\sqrt{4-{{x}^{2}}},\] \[y\ge \sqrt{2}\sin \left( \frac{x\pi }{2\sqrt{2}} \right)\], and the x-axis is divided by the y-axis in the ratio
  A)
  \[\frac{{{\pi }^{2}}-8}{{{\pi }^{2}}+8}\]        done clear
  B)
  \[\frac{{{\pi }^{2}}-4}{{{\pi }^{2}}+4}\] done clear
  C)
  \[\frac{\pi -4}{\pi -4}\]    done clear
  D)
  \[\frac{2{{\pi }^{2}}}{2\pi +{{\pi }^{2}}-8}\] done clear
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• question_answer8)
  Let\[f(x)={{x}^{3}}+3x+2\] and g(x) be the inverse of it. Then the area bounded by g(x), the x-axis, and the ordinate at \[x=-\,2\] and \[x=6\] is
  A)
  1/4 sq. units       done clear
  B)
  4/3 sq. units done clear
  C)
  5/4 sq. units       done clear
  D)
  7/3 sq. units done clear
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• question_answer9)
  The area bounded by the x-axis, the curve \[y=f(x)\], and the lines x = 1, x = b is equal to\[\sqrt{{{b}^{2}}+1}-\sqrt{2}\] for all b > l, then f(x) is
  A)
  \[\sqrt{x-1}\]      done clear
  B)
  \[\sqrt{x+1}\] done clear
  C)
  \[\sqrt{{{x}^{2}}+1}\] done clear
  D)
  \[\frac{x}{\sqrt{1+{{x}^{2}}}}\] done clear
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• question_answer10)
  The area enclosed between the curve \[{{y}^{2}}(2a-x)={{x}^{3}}\] and the line x = 2 above the x-axis is
  A)
  \[\pi {{a}^{2}}\]sq. units           done clear
  B)
  \[\frac{3\pi \,{{a}^{2}}}{2}\] sq. units done clear
  C)
  \[2\pi \,{{a}^{2}}\]sq. units done clear
  D)
  \[3\pi \,{{a}^{2}}\] sq. units done clear
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• question_answer11)
  Consider two curves \[{{C}_{1}}:{{y}^{2}}=4[\sqrt{y}]x\] and\[{{C}_{2}}:{{x}^{2}}=4[\sqrt{x}]y\], where [.] denotes the greatest integer function. Then the area of region enclosed by these two curves within the square formed by the lines x =1, y =1, x = 4, y = 4 is
  A)
  8/3 sq. units       done clear
  B)
  10/3 sq. units done clear
  C)
  11/3 sq. units     done clear
  D)
  11/4 sq. units done clear
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• question_answer12)
  The area bounded by the curve y = x \[\left| x \right|\], x-axis and the ordinates x = 1, \[x=-1\] is given by
  A)
  0          done clear
  B)
  1/3 done clear
  C)
  2/3                    done clear
  D)
  None of these done clear
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• question_answer13)
  Area lying in the first quadrant and bounded by the curve \[y={{x}^{3}}\] and the line y = 4x is
  A)
  2                      done clear
  B)
  3 done clear
  C)
  4          done clear
  D)
  5 done clear
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• question_answer14)
  The area bounded by the curves \[x={{y}^{2}}\] and \[x=\frac{2}{1+{{y}^{2}}}\] is
  A)
  \[\pi -\frac{2}{3}\] done clear
  B)
  \[\pi +\frac{2}{3}\] done clear
  C)
  \[-\pi -\frac{2}{3}\]         done clear
  D)
  none of these done clear
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• question_answer15)
  The area bounded by the curve y = sin x and the line x = 0, \[\left| y \right|=\frac{\pi }{2}\] is
  A)
  1          done clear
  B)
  2 done clear
  C)
  \[\pi \]    done clear
  D)
  2\[\pi \] done clear
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• question_answer16)
  The area of the region in 1st quadrant bounded by the y-axis, y = \[\frac{x}{4}\], y = 1 + \[\sqrt{x}\], and \[y=\frac{2}{\sqrt{x}}\] is
  A)
  2/3 sq. units       done clear
  B)
  8/3 sq. units done clear
  C)
  11/3 sq. units     done clear
  D)
  13/6 sq. units done clear
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• question_answer17)
  The area enclosed by the curves \[y=\text{sin}\,x+\text{cos}\,x\]and \[y=\left| \text{cos }x-\text{sin }x \right|\] over the interval [0,\[\pi /2\]] is
  A)
  \[4(\sqrt{2}-1)\]  done clear
  B)
  \[2\sqrt{2}(\sqrt{2}-1)\] done clear
  C)
  \[2(\sqrt{2}+1)\] done clear
  D)
  \[2\sqrt{2}(\sqrt{2}+1)\] done clear
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• question_answer18)
  The area enclosed by \[y={{x}^{2}}+\cos x\] and its normal at \[x=\frac{\pi }{2}\] in the first quadrant is
  A)
  \[\frac{{{\pi }^{5}}}{32}-\frac{{{\pi }^{4}}}{64}+\frac{{{\pi }^{3}}}{32}+1\] done clear
  B)
  \[\frac{{{\pi }^{5}}}{16}-\frac{{{\pi }^{4}}}{32}+\frac{{{\pi }^{3}}}{24}-1\] done clear
  C)
  \[\frac{{{\pi }^{5}}}{32}-\frac{{{\pi }^{4}}}{32}+\frac{{{\pi }^{3}}}{16}\] done clear
  D)
  \[\frac{{{\pi }^{5}}}{32}-\frac{{{\pi }^{4}}}{32}+\frac{{{\pi }^{3}}}{24}+1\] done clear
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• question_answer19)
  The area enclosed between the curves \[y=a{{x}^{2}}\] and \[x=a{{y}^{2}}\] (where a > 0) is 1 sq. unit, then the value of a is
  A)
  \[1/\sqrt{3}\]      done clear
  B)
  ½ done clear
  C)
  1                      done clear
  D)
  1/3 done clear
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• question_answer20)
  The area bounded by the curve \[{{y}^{2}}(2-x)={{x}^{3}}\] and x = 2 is
  A)
  \[\frac{\pi }{2}\]             done clear
  B)
  \[\pi \] done clear
  C)
  \[2\pi \]  done clear
  D)
  \[3\pi \] done clear
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• question_answer21)
  The area bounded by the curves \[y={{\log }_{e}}x,\,\,y={{\log }_{e}}\left| x \right|,\]\[y=\left| {{\log }_{e}}x \right|,\]and \[y=\left| {{\log }_{e}}\left| x \right| \right|\] is ____.
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• question_answer22)
  The area of the region bounded by the curves \[y=\left| x-2 \right|,\] x=1, x=3, and the x-axis is ____.
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• question_answer23)
  The area enclosed between the curve \[y={{\log }_{e}}(x+e)\] and the coordinate axes is ____.
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• question_answer24)
  The area of the region bounded by the curves \[y=\left| x-1 \right|\] and \[y=3-\left| x \right|\] is ____.
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• question_answer25)
  The area of the plane region bounded by the curves \[x+2{{y}^{2}}=0\] and \[x+3{{y}^{2}}=1\] is equal to _______.
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